Full-waveform inversion for reverse vertical seismic profiling data based on a variant of the optimal transport theory

Full-waveform inversion has become a useful tool to construct the details of the geological structure. However, due to the strong nonlinearity of the inverse problem and inaccuracy of the initial model, full-waveform inversion often faces severe cycle-skipping problems and thus may exhibit poor inversion results. The optimal transport-based full-waveform inversion can mitigate the cycle-skipping problem. For that, transformations of oscillatory seismic data to distributions need to satisfy positive and mass conservation conditions. In this work, we introduce a new data normalization method based on the Sigmoid function. The convexity of the objective function is improved compared to the traditional L-2 norm by setting appropriate parameters. Combined with the adaptive quadratic Wasserstein distance, our approach presents strong adaptability to a poor initial model and can reduce cycle skipping. Numerical examples are provided to validate the method in a reverse vertical seismic profiling geometry, and the inverted models obtained by different normalization methods are also compared.